#14927: Hyperbolic area sine is unstable for (even moderately) big negative arguments. -------------------------------------+------------------------------------- Reporter: leftaroundabout | Owner: (none) Type: bug | Status: new Priority: normal | Milestone: Component: libraries/base | Version: 8.2.1 Resolution: | Keywords: Operating System: Unknown/Multiple | Architecture: Type of failure: Incorrect result | Unknown/Multiple at runtime | Test Case: Blocked By: | Blocking: Related Tickets: | Differential Rev(s): Wiki Page: | -------------------------------------+------------------------------------- Comment (by leftaroundabout): Alternatives to the fix suggestion above: - {{{asinh x = signum x * log (abs x + sqrt (1 + x*x))}}} More concise form of the same thing; quite possibly faster performance because a branch can be avoided. - Actually, the Python version has also a higher ''upper'' limit: {{{ Prelude> asinh 1e200 In [3]: asinh(1e200) Infinity Out[3]: 461.2101657793691 }}} The reason is that `x*x` overflows even though the function as a whole could still be well-defined. I consider this not nearly as problematic as the much earlier problems I showed for the negative side, but it's still not great. This issue can be fixed by noting that for `x > 1/ε²`, we have anyways `1 + x*x == x*x` and thus `asinh x` gives the same result as `log (2*x) = log 2 + log x`. So, a ''really'' stable area hyperbolic sine would be {{{ asinh x | x > huge = log 2 + log x | x < 0 = -asinh (-x) | otherwise = log (1 + sqrt (1 + x*x)) }}} with {{{ huge = 1 / last (takeWhile ((>1).(+1)) $ recip . (2^) <$> [0..]) {- == 1 / Numeric.IEEE.epsilon -} }}} Mind, the exact value is completely uncritical, we could as well hard- code `huge = 1e20`. That implementation accurately left-inverts `sinh` (for `Double`) for all the range from `-710` to `710` (in other words, it right-inverts `sinh` for all the range from `-1.17e308` to `1.17e308`), which is basically the same as in Python. The current implementation only reaches up to `sinh 355`. -- Ticket URL: http://ghc.haskell.org/trac/ghc/ticket/14927#comment:1 GHC http://www.haskell.org/ghc/ The Glasgow Haskell Compiler